According to the poll, a decent amount of respondents — 43% — were able to identify correctly that "about 8 million people" had signed up through exchanges created by the health-care law. Obama trumpeted those final enrollment numbers at least three times: during a Rose Garden statement the day after the signup deadline, during a statement announcing the departure of Secretary of Health and Human Services Kathleen Sebelius, and during a press conference two weeks ago.

But based on the poll's findings, the American people aren't swayed by the strong enrollment numbers. Though respondents were largely aware of the numbers, a majority — 57% — still said the law's first enrollment period "fell short of expectations." Only 35% said it either met or exceeded expectations.

Here's a chart showing the contrast:

Kaiser Family Foundation

The number underscores that no matter how many people ultimately enrolled, Obamacare's tumultuous first enrollment period will be hard to tout ahead of this year's midterm elections where Democrats will be fighting to hold onto control of the Senate.

Overall, according to the Kaiser poll, the Affordable Care Act is viewed slightly more positively now than it was at the start of the year — 38% say they support the law, compared with 34% in January.

Another new poll released Tuesday from The Washington Post and ABC News also found the law could be a major hindrance for Democrats this fall. Among the findings of that poll, 47% of respondents said the law is making their health-care costs go up; just 7% said it's making costs go down and 41% say their health-care costs have stayed about the same.

Still, there's a silver lining for Democrats. Back-to-back recent polls showed more people are clearly lining up with Democrats' preferred position on Obamacare — keep the law and work to fix problems with it — than with Republicans' continued call for the law to be repealed and replaced.

According to the Kaiser poll, 58% of respondents said Congress should work to improve the law. Only 35% said they think Congress should continue to work toward repealing and replacing the law at this point. Those results align with findings from a Democracy Corps poll released Monday, which dug a bit deeper into individual states' views on the subject.

In December, according to the poll, 47% of voters in Republican-held districts said they wanted to "implement and fix" the law, compared with 46% who were for "repeal and replace." Those numbers now: 53% for implementing and fixing, and only 42% for repealing and replacing.